A Method of Rotations for Lévy multipliers

Michael Perlmutter

doi:10.1007/s00209-017-1845-8

A Method of Rotations for Lévy multipliers

Michael Perlmutter

Purdue University

Research output: Contribution to journal › Article › peer-review

Abstract

We use a method of rotations to study the Lp boundedness, for 1 < p < ∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric α -stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2, and therefore allows us to obtain a larger class of multipliers which are bounded on Lp . As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Lp boundedness of the Beurling-Ahlfors transform.

Original language	American English
Journal	Mathematische Zeitschrift
Volume	287
Issue number	3-4
DOIs	https://doi.org/10.1007/s00209-017-1845-8
State	Published - Dec 2017
Externally published	Yes

Keywords

Beurling–Ahlfors transform
Fourier multipliers
Lévy processes
martingale transforms
method of rotations

EGS Disciplines

Mathematics

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10.1007/s00209-017-1845-8License: Unspecified

Cite this

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title = "A Method of Rotations for L{\'e}vy multipliers",

abstract = " We use a method of rotations to study the Lp boundedness, for 1 < p < ∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric α -stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2, and therefore allows us to obtain a larger class of multipliers which are bounded on Lp . As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Lp boundedness of the Beurling-Ahlfors transform.",

keywords = "Beurling–Ahlfors transform, Fourier multipliers, L{\'e}vy processes, martingale transforms, method of rotations",

author = "Michael Perlmutter",

note = "We use a method of rotations to study the \$\$L\textasciicircum{}p\$\$ L p boundedness, for \$\$1<p<\textbackslash{}infty \$\$ 1 < p < ∞ , of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric \$\$\textbackslash{}alpha \$\$ α -stable processes, \$\$0<\textbackslash{}alpha <2\$\$ 0 < α < 2 .",

year = "2017",

month = dec,

doi = "10.1007/s00209-017-1845-8",

language = "American English",

volume = "287",

number = "3-4",

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TY - JOUR

T1 - A Method of Rotations for Lévy multipliers

AU - Perlmutter, Michael

N1 - We use a method of rotations to study the $$L^p$$ L p boundedness, for $$1<p<\infty $$ 1 < p < ∞ , of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric $$\alpha $$ α -stable processes, $$0<\alpha <2$$ 0 < α < 2 .

PY - 2017/12

Y1 - 2017/12

N2 - We use a method of rotations to study the Lp boundedness, for 1 < p < ∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric α -stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2, and therefore allows us to obtain a larger class of multipliers which are bounded on Lp . As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Lp boundedness of the Beurling-Ahlfors transform.

AB - We use a method of rotations to study the Lp boundedness, for 1 < p < ∞, of Fourier multipliers which arise as the projection of martingale transforms with respect to symmetric α -stable processes, 0 < α < 2. Our proof does not use the fact that 0 < α < 2, and therefore allows us to obtain a larger class of multipliers which are bounded on Lp . As in the case of the multipliers which arise as the projection of martingale transforms, these new multipliers also have potential applications to the study of the Lp boundedness of the Beurling-Ahlfors transform.

KW - Beurling–Ahlfors transform

KW - Fourier multipliers

KW - Lévy processes

KW - martingale transforms

KW - method of rotations

UR - https://doi.org/10.1007/s00209-017-1845-8

U2 - 10.1007/s00209-017-1845-8

DO - 10.1007/s00209-017-1845-8

M3 - Article

VL - 287

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -

A Method of Rotations for Lévy multipliers

Abstract

Keywords

EGS Disciplines

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